An injectivity theorem on snc compact K\"ahler spaces: an application of the theory of harmonic integrals on log-canonical centers via adjoint ideal sheaves

Abstract

Let \((X,D)\) be a log-canonical (lc) pair, in which \(X\) is a compact K\"ahler manifold and \(D\) is a reduced snc divisor, and let \(F\) be a holomorphic line bundle on \(X\) equipped with a smooth metric \(h_F = e^{-\varphi_F}\). Via the use of the adjoint ideal sheaves (constructed from \(\varphi_F\) and \(D\)) and the associated residue morphisms, sections of \(K_D \otimes \left. F\right|_D\) on \(D\) (as well as those of \(K_X \otimes D \otimes F\) on \(X\)) can be related to the \(F\)-valued holomorphic top-forms on each lc center of \((X,D)\) by an inductive use of a certain residue exact sequence derived from the adjoint ideal sheaves. The theory of harmonic integrals is valid on each lc center (which is compact K\"ahler), so this provides a pathway to apply the techniques in harmonic theory to the possibly singular K\"ahler space \(D\). To illustrate the use of such apparatus in problems concerning lc pairs, we prove a Koll\'ar-type injectivity theorem for the cohomology on \(D\) when \(F\) is semi-positive. This in turn also solves the conjecture by Fujino on the injectivity theorem for the compact K\"ahler lc pair \((X,D)\), providing an alternative proof of a recent result by Cao and P\u{a}un.